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Mirrors > Home > ILE Home > Th. List > iccsupr | Unicode version |
Description: A nonempty subset of a closed real interval satisfies the conditions for the existence of its supremum. To be useful without excluded middle, we'll probably need to change not equal to apart, and perhaps make other changes, but the theorem does hold as stated here. (Contributed by Paul Chapman, 21-Jan-2008.) |
Ref | Expression |
---|---|
iccsupr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccssre 8978 | . . . 4 | |
2 | sstr 3007 | . . . . 5 | |
3 | 2 | ancoms 264 | . . . 4 |
4 | 1, 3 | sylan 277 | . . 3 |
5 | 4 | 3adant3 958 | . 2 |
6 | ne0i 3257 | . . 3 | |
7 | 6 | 3ad2ant3 961 | . 2 |
8 | simplr 496 | . . . 4 | |
9 | ssel 2993 | . . . . . . . 8 | |
10 | elicc2 8961 | . . . . . . . . 9 | |
11 | 10 | biimpd 142 | . . . . . . . 8 |
12 | 9, 11 | sylan9r 402 | . . . . . . 7 |
13 | 12 | imp 122 | . . . . . 6 |
14 | 13 | simp3d 952 | . . . . 5 |
15 | 14 | ralrimiva 2434 | . . . 4 |
16 | breq2 3789 | . . . . . 6 | |
17 | 16 | ralbidv 2368 | . . . . 5 |
18 | 17 | rspcev 2701 | . . . 4 |
19 | 8, 15, 18 | syl2anc 403 | . . 3 |
20 | 19 | 3adant3 958 | . 2 |
21 | 5, 7, 20 | 3jca 1118 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 w3a 919 wceq 1284 wcel 1433 wne 2245 wral 2348 wrex 2349 wss 2973 c0 3251 class class class wbr 3785 (class class class)co 5532 cr 6980 cle 7154 cicc 8914 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-po 4051 df-iso 4052 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-icc 8918 |
This theorem is referenced by: (None) |
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